[Paper Review] A classification result on prime Hopf algebras of GK-dimension one

Gongxiang Liu|arXiv (Cornell University)|Apr 24, 2018

Algebraic structures and combinatorial models23 references3 citations

TL;DR

This paper classifies prime Hopf algebras of Gelfand-Kirillov dimension one under two conditions: existence of a 1-dimensional representation of order equal to the PI-degree of the algebra, and invariance of the components under this representation. The classification yields new non-pointed Hopf algebras of GK-dimension one, including two new series of finite-dimensional Hopf algebras, one of which includes a Hopf algebra of dimension 24, and provides a partial answer to a question posed in [9].

ABSTRACT

In this paper, we classify all prime Hopf algebras $H$ of GK-dimension one satisfying the following two conditions: 1) $H$ has a 1-dimensional representation of order PI.deg$(H)$ and 2) the invariant components of $H$ with respect to this 1-dimensional representation are domains (see Section 2 for related definitions). As consequences, 1) a number of new Hopf algebras of GK-dimension one are found and some of them are not pointed, 2) we give a partial answer to a question posed by Brown and Zhang and 3) some new kinds of finite-dimensional Hopf algebras are found.

Motivation & Objective

• To classify prime Hopf algebras of GK-dimension one satisfying two structural conditions: existence of a 1-dimensional representation of order equal to the PI-degree and invariant components that are domains.
• To extend the classification of prime regular Hopf algebras of GK-dimension one by relaxing the regularity assumption.
• To identify new examples of non-pointed Hopf algebras of GK-dimension one, which are rare in the literature.
• To provide a partial answer to a question posed in [9] regarding the structure of prime regular Hopf algebras of GK-dimension one.
• To construct two new families of finite-dimensional Hopf algebras, including one with a 24-dimensional example.

Proposed method

• Utilizes the framework of Yetter-Drinfeld modules over group algebras $\mathbbm{k}\mathbb{Z}_n$ to realize singular curves as Hopf algebras in braided tensor categories.
• Applies the theory of Nichols algebras and braided Hopf algebras to analyze the structure of prime Hopf algebras of GK-dimension one.
• Employs the concept of invariant components under a 1-dimensional representation to constrain the algebraic structure.
• Uses graded algebra techniques and monomial regularity arguments to prove primality of certain algebras, such as $\Lambda(n)$, by showing that nontrivial ideals must contain monomials.
• Applies the PI-degree formula $\text{PI-deg}(\Lambda(n)) = 2^{m+1}$ for $\Lambda(n)$ with $n = 2(m+1)$ to verify the 1-dimensional representation condition.
• Analyzes the structure of the left invariant component under a 1D representation and shows it fails to be a domain unless commutative, leading to a key obstruction in Hypothesis (Hyp2).

Experimental results

Research questions

• RQ1 Can singular irreducible curves, such as the cusp $y_1^2 = y_2^3$, be realized as Hopf algebras in the braided category ${}^{\mathbb{Z}_n}_{\mathbb{Z}_n}\mathcal{YD}$?
• RQ2 Does every prime Hopf algebra of GK-dimension one with a 1-dimensional representation of order equal to its PI-degree and domain invariant components admit a classification?
• RQ3 Is the existence of a 1-dimensional representation with domain invariant components (Hyp2) equivalent to the existence of such a representation with order equal to the PI-degree (Hyp1)?
• RQ4 Can new non-pointed Hopf algebras of GK-dimension one be constructed beyond the known examples?
• RQ5 Does every prime Hopf algebra of GK-dimension one fit into an exact sequence of the form $\mathbbm{k} \to \text{alg.gp} \to H \to \text{f.d. Hopf} \to \mathbbm{k}$?

Key findings

• The paper constructs a new class of prime Hopf algebras of GK-dimension one that are not pointed, including a 24-dimensional finite-dimensional Hopf algebra.
• It identifies a new example of a Hopf algebra structure on the cusp $y_1^2 = y_2^3$ realized in the braided category ${}^{\mathbb{Z}_6}_{\mathbb{Z}_6}\mathcal{YD}$.
• The PI-degree of the algebra $\Lambda(n)$ is computed as $2^{m+1}$ when $n = 2(m+1)$, confirming the 1D representation condition in the classification.
• The classification result shows that the invariant components under the 1D representation are domains only under specific structural constraints, and such components are not domains in general.
• The paper provides a partial answer to a question in [9] by showing that the classification of prime regular Hopf algebras of GK-dimension one can be extended to non-regular cases under the given hypotheses.
• The conjecture is confirmed to hold for all examples in the paper: every such prime Hopf algebra of GK-dimension one is commutative-by-finite, i.e., finite over a commutative normal Hopf subalgebra.